The phenomenological validation is particularly important given that
our method does introduce model-dependent input in the intermediate-momentum region, albeit
anchored to known constraints from QCD at low and high momenta. The extraction of the LO contact term, in a given scheme, then proceeds
by matching the full amplitude obtained in this way to the LO chiral EFT amplitude. We considered several sources of uncertainty,
chief among them the missing contributions from inelastic intermediate states, such as \(N\!N \pi \) .
The intermediate steps of our analysis have been performed in dimensional regularization with the \(\overline{\text{MS}}\)  scheme, while the final result can be expressed in terms of the scheme-independent renormalized amplitude \(\mathcal {A}_\nu (|\mathbf {p}|,|\mathbf {p}^\prime |)\) 
at a set of kinematic points near threshold, where LO chiral EFT is expected to give an excellent approximation.
Using our synthetic data, as discussed in Section , one can then determine the contact term in any
regularization and renormalization scheme, in particular the ones employed in nuclear-structure calculations
for isotopes of experimental interest for \(0\nu \beta \beta \)  searches. This application is timely in view of the remarkable progress in ab-initio calculations of \(0\nu \beta \beta \)  decay rates of light and intermediate-mass nuclei [1]}, [2]}, [3]}, [4]}, [5]}, ranging from \({}^6\) He to \({}^{48}\) Ca and \({}^{76}\) Ge, starting from microscopic nuclear forces obtained from chiral EFT. So far these decay rates include the long-distance neutrino-exchange contributions while omitting the contact term, which can now be remedied using our synthetic data for \(nn\rightarrow pp e^-e^-\) , allowing, for the first time, for
complete LO calculations of nuclear \(0\nu \beta \beta \)  decay rates.
For heavier nuclei such as \({}^{136}\) Xe, which are still beyond the capabilities of ab-initio techniques, the impact of the short-range term could be studied indirectly, e.g., by matching ab-initio results and nuclear models for nuclei accessible to both approaches (this strategy was used in Ref. [6]} for the axial-vector current).
